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A worked example provides a step-by-step solution to a problem. The following is an example from algebra:

Make a the subject of the equation, + =( ) / .a b c d

Solution

( )/+ =a b c d+ =a b dc

= −a dc b

Learners can be presented this worked example to study. Alternatively, they can be asked to solve a problem. Learners asked to solve a problem are just presented the first line of the above worked example, ‘Make a the subject of the equation,

,( )/+ =a b c d . The worked example effect occurs when learners presented worked examples to study perform better on subsequent test problems than learners asked to solve the equivalent problem.

The worked example effect flows directly from the cognitive architecture dis-cussed in the initial parts of this book. Worked examples can efficiently provide us with the problem-solving schemas that need to be stored in long-term memory using the information store principle. Once stored in long-term memory, we can use the stored schemas to solve related problems using the environmental organising and linking principle. Those schemas are borrowed from the long-term memory of the provider of the worked example by way of the borrowing and reorganising principle. Worked examples impose a relatively low working memory load (narrow limits of change principle) compared to solving problems using means–ends search. While all the necessary, intrinsic interacting elements are encapsulated in the information contained within a worked example, solving a problem by means–ends search adds the additional elements associated with the randomness as genesis principle. That principle unnecessarily adds problem-solving search to the interacting elements, thus imposing an extraneous cognitive load. Together, these various mechanisms of cognitive load theory suggest that for novice learners, studying worked examples should be superior to solving the equivalent problems. There is a wealth of evidence supporting this hypothesis collected by researchers from around the globe.

Chapter 8The Worked Example and Problem Completion Effects

J. Sweller et al., Cognitive Load Theory, Explorations in the Learning Sciences, Instructional Systems and Performance Technologies 1,DOI 10.1007/978-1-4419-8126-4_8, © Springer Science+Business Media, LLC 2011

100 8 The Worked Example and Problem Completion Effects

Atkinson, Derry, Renkl, and Wortham (2000) observed there is no precise definition of a worked example but there are a number of common features found across the different types. Most worked examples include a problem statement and procedure for solving the problem. By studying a worked example, students are able to learn key aspects about the problem and use those aspects to solve other problems. As Atkinson et al. (2000) further remark ‘In a sense, they provide an expert’s problem-solving model for the learner to study and emulate’ (p. 181). A number of different synonyms have been applied to worked examples such as learning from examples, example-based learning, learning from model answers and studying expert solutions.

Basic Empirical Evidence

Worked examples are not a recent innovation as teachers, particularly in mathemat-ics and science, have used them extensively over a long period of time. However, as will become evident below, there are optimal ways of presenting worked examples. A traditional textbook approach in mathematics and science, when learning new concepts and procedures, is to present some initial worked examples and then ask students to practice what they have learned in a more extensive exercise including unfamiliar problems. Typically, under some conditions, students may only be shown further worked examples after failing to find solutions to some of the problems. Frequently, worked examples are not even used when students fail to solve a prob-lem. Under such circumstances, learners are still required to spend a significant time on problem solving. As described in Chapter 7, problem solving via means–ends analysis requires problem solvers to process a large number of interacting elements and can create an extraneous cognitive load that inhibits learning. Even though some exposure to worked examples is used in most traditional instructional procedures, worked examples, to be most effective, need to be used much more systematically and consistently to reduce the influence of extraneous problem-solving demands.

Research into worked examples has a long history. Atkinson et al. (2000) reported that as far back as the 1950s, researchers used learning-by-example strate-gies to investigate the processes involved in concept formation. Whereas cognitive load theory researchers have also focused on concept or schema formation, many of their studies have explicitly compared worked example approaches to learning with a problem-solving approach. These comparisons have led to the identification of the worked example effect.

Worked Examples in Mathematics and Related Domains

Early evidence for the worked example effect came from studies involving the learning of mathematics. Sweller and Cooper (1985) used algebraic manipulation problems (e.g. for the equation a = af + c, express a in terms of the other variables) to show that worked examples required less time to process than solving the equivalent,

101Basic Empirical Evidence

conventional problems during acquisition, and led to quicker solutions times and lower error rates on similar test problems. The experimental design used in this study, which became a blueprint for many following studies, directly compared a worked example group with a conventional problem-solving group. Initially, both groups of high school students (Year 9) were presented a limited number of worked examples of the new material to be learned, in this case, solutions to alge-bra manipulation problems. This introductory phase was followed by the main learning acquisition phase. For the worked example group, students were presented a set of problem pairs consisting of a worked example to study and then immedi-ately after, a similar problem to solve. This example–problem pair format was repeated several times with different problems to form the acquisition problem set. The conventional group was presented the same problem set but was required to solve all the problems, as students were not given any worked examples to study during this phase. In this design, the worked example group was asked to solve half the number of problems that the conventional group had to solve, but was also asked to study worked solutions to the other half. Following acquisition, both groups were presented a set of test problems to solve without the inclusion of any worked examples.

While Sweller and Cooper (1985) found improved test performance by the worked example group on problems similar to the acquisition problems, they failed to find evidence of transfer. The worked example group did not have an advantage over the conventional group on dissimilar problems. In a follow up study, Cooper and Sweller (1987) set out to investigate the conditions under which worked examples could facilitate transfer. They ran a series of experiments using both algebra manip-ulation problems and word problems to test the hypothesis that in order for transfer to take place, automation of problem-solving operators is necessary.

Rule or schema automation allows a procedure to be used with minimal working memory resources (Chapter 2). For example, we may be able to multiply out a denominator in a fractional algebraic equation automatically without actively think-ing about the process. In contrast, when first learning to multiply out a denomina-tor, we may need to consider the process every time we use it. Automation means working memory resources are available for other activities during problem solving. If we are presented with a novel problem that requires a denominator to be multi-plied out, we can devote working memory resources to finding a solution rather than attempting to recall how the relevant rule works. In this manner, if worked examples facilitate automation more than solving the equivalent problems, transfer should be facilitated resulting in transfer effects.

However, automation takes place slowly and therefore requires substantial acquisition time, which the previous Sweller and Cooper (1985) study did not pro-vide. In contrast, the Cooper and Sweller (1987) experiments provided extra learn-ing time, enabling the worked example group to demonstrate significant transfer effects compared with the conventional group. Cooper and Sweller concluded that in any complex domain, significant acquisition time is required to automate the required problem-solving operators to demonstrate transfer. Worked examples were found to accelerate this process compared with a problem-solving approach.

102 8 The Worked Example and Problem Completion Effects

A later study by Carroll (1994) found that worked examples were particularly helpful for students with a history of low achievement in mathematics and those identified as learning disabled. Pillay (1994) extended the research into the use of worked examples in mathematics by showing the advantage of using worked examples over problem solving when learning 2D and 3D mental rotations. Paas (1992) using statistics problems and Paas and van Merriënboer (1994) using geom-etry problems found strong evidence for the worked example effect. Paas’ (1992) work will be discussed in more detail when discussing the completion effect in this chapter while Paas and van Merriënboer’s (1994) work will be discussed further in Chapter 16 when discussing the variability effect.

Worked Examples and Ill-Structured Learning Domains

The bulk of research on the worked example effect has used well-structured prob-lems from mathematics or science domains rather than ill-structured problems requiring natural language, humanities or other areas related to artistic endeavours. A well-structured problem is one in which we can clearly specify the various prob-lem states and the problem-solving operators (e.g. the rules of algebra) required to move from one state to another. Ill-structured problems do not have clearly speci-fied problem states or problem-solving operators. ‘Discuss the meaning of this passage’ provides an example of an ill-structured problem.

It has been suggested by some (e.g. Spiro & DeSchryver, 2009) that the worked example effect cannot be obtained using ill-structured problems. In fact there are theoretical reasons to suppose that the cognitive activities involved in both solving and learning to solve ill-structured problems are identical to those required to solve well-structured problems (Greeno, 1976). The cognitive architecture discussed in the previous parts does not distinguish between well-structured and ill-structured problems and there is no reason to suppose we have a different architecture to deal these differing categories of problems. We must acquire schematically based knowledge that allows us to recognise problem types and the categories of solution moves to solve particular categories of problems irrespective of whether the prob-lems are well structured or ill structured. The solution variations available for ill-structured problems are larger than for well-structured problems but they are not infinite and experts have learned more of the possible variations than novices. Of course, the ultimate test is whether the worked example effect can be obtained using ill-structured problems.

In a review of worked examples, Renkl (2005) made a number of insightful comments and recommendations on future research into worked examples. One point made was that they could be ‘relevant only to a limited range of domains’ (p. 241). Renkl argued that they seem to be particularly suited to skill domains where algorithms can be applied, i.e. well-structured problems. He commented further that in areas such as writing a text or interpreting a poem, the essential strength of worked examples in showing solution steps may not be present. As indicated above,

103Basic Empirical Evidence

the vast majority of the experiments described so far in this chapter have used algorithmic-based domains such as mathematics, science and computing. Nevertheless, in recent years, an increasing amount of research has been conducted within a cognitive load theory framework in more ill-defined domains. For example although not directly testing the worked example effect (there were no problem-solving groups), Owens and Sweller (2008) demonstrated that music instruction could be effectively formatted using worked examples. Similarly Diao and Sweller (2007) and Diao, Chandler, and Sweller (2007) used worked examples in the domain of second language learning.

Three studies of particular note have demonstrated the worked example effect in ill-defined problem areas. Firstly, Rourke and Sweller (2009) required university students to learn to recognise particular designers’ styles from the early Modernist period using chair designs. It was found that a worked example approach was supe-rior to problem solving in recognising these designs. Furthermore the worked example effect extended to transfer tasks in the form of other designs, based on stained glass windows and cutlery.

Secondly, in two experiments, Oksa, Kalyuga, and Chandler (2010) presented novices (Grade 10 students) with extracts from Shakespearean plays. One group was given explanatory notes integrated into the original text, whereas a second group had no such notes. Results indicated that the explanatory notes group outper-formed the unsupported group on a comprehension task and reported a lower cogni-tive load. The design of the Oksa et al. experiments does not fit the traditional worked example alternation format of study–solve problem pairs because these experiments were part of a wider study on the expertise reversal effect (see Chapter 12). Nevertheless, half of the students were provided model answers or interpretations to key aspects of the text. Those model answers are equivalent to problem solutions. In contrast, students with no explanatory notes were required to make their own interpretations, an activity equivalent to problem solving. The fact that the model answers resulted in more learning than requiring students to make their own inter-pretations in this very ill-structured domain provides strong evidence that the worked example effect is applicable to ill-structured problem domains.

Thirdly, Kyun, Kalyuga, and Sweller (in preparation) also demonstrated the worked example effect in learning English literature. More- and less-knowledgeable Korean university students for whom English was a foreign language were used in another study on the expertise reversal effect (Chapter 12). During the learning phase, half of the students were presented conventional essay questions that they were asked to answer. The other half of the students were presented the same ques-tions along with model answers that they were asked to study, followed by similar questions that they had to answer themselves. All students then were asked to answer retention, near and far transfer tests. The less-knowledgeable students in the worked example group were rated by markers as having performed better on the problems of the learning phase. Even though there were no significant effects for transfer tests, for the retention test, the worked example group performed signifi-cantly better than the conventional problem-solving group. Again, the worked example effect was demonstrated using an ill-structured problem.

104 8 The Worked Example and Problem Completion Effects

Worked Examples in Non-Laboratory-Based Experiments

The early research into worked examples described above was conducted under controlled laboratory-style conditions. Evidence also emerged that a worked exam-ples approach could be implemented effectively on a much wider scale and under everyday classroom conditions. In a longitudinal study with Chinese students, Zhu and Simon (1987) showed that worked examples could be successfully substituted for lectures and other traditional mathematics classroom activities over a prolonged period. They found that a mathematics course that was traditionally taught in 3 years could be completed in 2 years with enhanced performance using a compre-hensive strategy based on worked examples.

Although not as extensive as the Zhu and Simon study, other studies have also been conducted within realistic learning settings. For example, a critical aspect of the Ward and Sweller (1990) study was that students studied worked examples dur-ing homework as part of a normal class. Carroll (1994) also administered a similar homework procedure. In both cases, a worked example effect was found.

Worked Examples and the Alternation Strategy

In most of the studies described above worked examples were presented in an example–problem pair format. During acquisition, pairs of similar problems were presented as the main learning vehicle. This methodology of pairing by studying a worked example and solving a similar problem was first adopted by Sweller and Cooper (1985). They created this alternation strategy on motivational grounds. It was assumed that students would be more motivated to study the worked example if they knew that they had to solve a similar problem immediately afterwards. Sweller and Cooper were concerned initially that students would not necessarily process the information in a worked example at a sufficient depth to assist schema acquisition if each worked example was not followed immediately by a similar problem to solve, and therefore created the study–solve strategy, that other researchers adopted.

To test the effectiveness of the alternation strategy, Trafton and Reiser (1993) completed a study that included blocked practice and alternating practice. Two types of blocked practice were included: study a set of several examples and then solve a similar set of problems, or solve a set of problems and then solve a set of similar problems. In addition, there were two types of alternating practice: Study an example and then immediately solve a similar problem, or solve a problem and immediately solve a second similar problem. Trafton and Reiser found that for an example to be most effective, it had to be accompanied by a problem to solve. The most efficient method of studying examples and solving problems was to present a worked example and then immediately follow this example by asking the learner to solve a similar problem. This efficient technique was, in fact, identical to the method used by Sweller and Cooper (1985) and followed in many other studies.

105The Problem Completion Effect

It was notable that the method of showing students a set of worked examples followed later by a similar set of problems to solve led to the worst learning outcomes.

The Problem Completion Effect

The worked example effect is related to several other important instructional effects. Some of those effects were discovered while studying the worked example effect. The split-attention, redundancy, modality, expertise reversal, guidance fad-ing and variability effects will be discussed in other chapters but the problem completion effect will be discussed in this chapter because it is closely related to the worked example effect.

One early concern about the use of worked examples was that they led to passive rather than more active learning. Would learners attend to and study the worked examples in enough depth or would they simply gloss over them? Furthermore, evidence had emerged that students may only study worked examples in depth if they find difficulty in solving conventional problems (Chi, Bassok, Lewis, Reimann, & Glaser, 1989). These issues suggest that learners may need to know that they have a similar problem to solve in order to fully process the example. As previously discussed, the paired alternation strategy (study an example–solve a problem) was developed to address this issue. Another strategy to ensure learners paid sufficient attention to the worked examples was to provide learners with completion problems (van Merriënboer & Krammer 1987). A completion problem is a partial worked example where the learner has to complete some key solution steps. The algebra worked example presented at the beginning of this chapter can be converted to a completion problem by only demonstrating the first step and then requiring learners to work out the second step themselves, as the following example indicates:

Make a the subject of the equation, + =( ) / .a b c d

Solution

( ) /a b c d+ =+ =a b dc

?=a

Van Merriënboer (1990) conducted the first extensive study on completion problems within a cognitive load theory paradigm using an introductory computer- programming course. Over a period of ten lessons, students followed either a conven-tional strategy in which they were asked to design and code new computer programs or a completion strategy that required the modification and extension of existing computer programs. It was found that the completion group was superior at subse-quently constructing new programs, providing an example of the completion effect.

Van Merriënboer and de Croock (1992) also conducted a similar study with computer-programming content. A generation (conventional) group was compared with a com-pletion group in learning about programming techniques. Results indicated superior

106 8 The Worked Example and Problem Completion Effects

learning by the completion group. When using a completion strategy, the presentation of new information and programming practice were linked to incomplete programs and learners were only required to complete the partial solutions, whereas the genera-tion strategy presented both model programs and generation assignments. Although the model programs could be considered as worked examples, the experimental design was such that students did not necessarily need to study them immediately. Data indi-cated that the conventional group frequently had to search for examples while solving their problems. In this learning domain, computer programs are very complex; conse-quently asking students to generate new programs may cause a high working memory load, which is intensified by the need for learners to search for, and refer back to, equally complex model programs, thus creating a high extraneous cognitive load. Extraneous cognitive load is reduced by presenting learners with appropriate worked examples prior to problem solving so that they do not have to search for examples while problem solving. These two studies (van Merriënboer, 1990; van Merriënboer & de Croock, 1992) illustrated that worked examples that have many solution steps may themselves generate additional extraneous load, but it can be offset by using comple-tion problems.

Paas (1992) expanded this research by comparing three groups (conventional problem solving, worked example and completion problems) in learning about elementary statistical concepts. Results indicated that both the worked example and completion groups had superior outcomes to the conventional group on both near and far transfer tasks, and also required less mental effort. A later study by van Merriënboer, Schuurman, de Croock, and Paas (2002a) found that completion prob-lem superiority may be limited to far transfer effects, although it was also demon-strated that a conventional condition created more cognitive load and was less efficient than a completion approach.

To explain the effectiveness of completion problems, Sweller (1999) argued that the inclusion of an element of problem solving could ensure that learners consider the problem in sufficient depth to attend to key information. By avoiding full problem solving, working memory is not overloaded. In order to complete the problem, the learner must attend to and process the worked-out part and then respond to the incomplete steps. Completion problems are a hybrid, including ele-ments of both a worked example and a problem to be solved (Clark, Nguyen, & Sweller, 2006). The completion effect was the first alternative to the standard for-mat for worked examples.

Critiques of the Use of Worked Examples

The worked example effect has been criticised with the suggestion that it only is obtained because of the use of an inappropriate control group (Koedinger & Aleven, 2007). Under problem-solving conditions testing for the worked example effect, students usually are asked to solve problems without any kind of support. In contrast, computer-based, problem-solving tutors frequently provide support when

107Conditions of Applicability

learners fail to solve a problem by indicating appropriate steps. In this manner, an equivalent of worked example steps may be presented after or during problem-solving failure rather than prior to a problem being presented for solution. It can be argued that this type of supported problem solving may constitute a more appropri-ate control group for worked examples.

It is true that most studies demonstrating the worked example effect have used problem-solving control groups with little or no support. Learners were simply required to solve problems after a limited introduction to a new topic with no support while solving the problem. In defence of this procedure, these conditions mirror common practices in educational institutions as well as being recom-mended by problem-solving advocates. One merely has to inspect any commonly used textbook in mathematics or science and peruse the long lists of problems presented to learners with minimal numbers of worked examples to obtain an indication of commonly used procedures. Students are required to learn by solving large numbers of problems.

As it happens, even if problem solving is supported in a computer-based envi-ronment, studying worked examples is still superior. Schwonke, Renkl, Krieg, Wittwer, Aleven, and Salden (2009) found that the worked example effect was still present in a well-supported problem-solving domain using a computer-based, cog-nitive tutor. Studying worked examples provides one of the best, possibly the best, means of learning how to solve problems in a novel domain.

Other criticisms of the use of worked examples have tended to be more ideologi-cally driven. Constructivists in particular tend to consider worked examples to be a form of knowledge transmission, devoid of active learning and devoid of much-valued problem-solving experience. Of course, whether learning is active is unre-lated to the physical activity of learners. One can be just as mentally active when studying a worked example as when solving a problem. It is the cognitive conse-quences of the activity that matters. Based on the borrowing and reorganising principle (see Chapter 3), activity that results in the acquisition of information from others is a very efficient way of learning and should not be down-played. Worked examples reduce extraneous cognitive load and can substantially increase the effec-tiveness of learning. As Kirschner, Sweller, and Clark (2006) pointed out, the use of discovery learning and problem solving during learning have a very weak research and theoretical base in contrast to the use of worked examples.

Conditions of Applicability

There is substantial evidence that learners, particularly those in the initial stages of cognitive skill acquisition, benefit more from studying worked examples than an equivalent episode of problem solving. Nevertheless, we need to remember that worked examples are effective because they reduce extraneous cognitive load. It is all too easy to assume that worked examples are effective because they are worked examples. A badly structured worked example presented to learners may be no

108 8 The Worked Example and Problem Completion Effects

more effective or even less effective than solving the equivalent problem. If extraneous cognitive load is not reduced compared to problem solving, the use of worked examples will not be effective.

The conditions under which worked examples are effective depend on the char-acteristics of the material and the characteristics of the learner. All of the effects discussed in the following chapters are concerned with the manner in which instruction should be presented to particular categories of learners. Many of those effects apply directly to the use of worked examples as well as other forms of instruction. In particular, the split-attention, modality, redundancy, expertise rever-sal, guidance fading, element interactivity, self-explanation and imagination effects apply to worked examples as well as other forms of instruction. The factors associ-ated with these effects all need to be considered when constructing worked exam-ples and will be outlined in the following chapters.

Instructional Implications

The research on the worked example effect has some very clear implications for instruction. Asking students to problem solve, particularly those learning new con-cepts and procedures (novices in the domain), creates an extraneous cognitive load that is detrimental to learning. Instead there should be a systematic process of using worked examples in the sense that worked examples should be programmed to include the alternation strategy (or a guidance fading strategy discussed in Chapter 13) and consist of extensive practice prior to solving sets of problems unaided.

Critics of cognitive load theory have tended to treat worked examples as a form of passive learning. Studying worked examples can be passive but passivity can be easily avoided. The use of example–problem pairs provides a simple technique that avoids passive learning, as do completion problems and guidance fading. As the weight of research evidence has become more compelling, worked examples have become more prominent in education communities. For example, a recent US Department of Education document recommended an emphasis on worked exam-ples (Pashler et al., 2007).

Conclusions

In over 25 years of cognitive load theory–based research the worked example effect has been shown to be very robust. Compelling evidence indicates that learners have a decided advantage in studying worked examples rather than solving equivalent problems.

Arguably, the worked example effect is the most important of the cognitive load theory effects. It has certainly been the most widely investigated. While the effect originated from cognitive load theory, the theory itself subsequently has been

109Conclusions

influenced by findings associated with the comparison of studying worked examples or solving problems. For example, the emphasis on the borrowing and reorganising principle in the current, evolutionary version of cognitive load theory relies heavily on the existence of the worked example effect. Furthermore, the effect can be difficult to explain by theories that place an emphasis on discovering or constructing information as opposed to obtaining that information from instructors (Kirschner, Sweller, & Clark, 2006).

The worked example effect has given rise to many other cognitive load theory effects, discussed in some of the subsequent chapters. The next chapter considers the split-attention effect, an effect that is critical to the effectiveness of worked examples.